Adjunction between a comonad and a monad

Question

Although this looks like elementary, I have trouble understanding the proof of Theorem 3.1 at page 7 of this paper. As hypotheses we are given a comonad $D$, a monad $T$ and an adjunction $D \dashv T$. But then in the course of the proof (at the top of page 8), the author constructs the monad $T$ from the comonad $D$. Why is the constructed monad identical to the one in hypotheses?

Answer 1

This all follows from the calculus of mates, which gives a natural bijection between natural transformations $\alpha:F\to F'$ between left adjoints and natural transformations $G'\to G$ between their right adjoints. Specifically, the mate $\bar{\alpha}$ is defined as the composite $$G'\stackrel{\eta_{G'}}{\to}GFG'\stackrel{G\alpha G'}{\to}GF'G'\stackrel{G\varepsilon'}{\to}G$$ where $\eta$ is the unit of $F\vdash G$ and $\varepsilon'$ is the counit of $F'\vdash G'$.

More abstractly, the mate construction may be interpreted as giving an equivalence between the 2-categories $\mathbf{Cat}_{\mathrm{Radj}}$ and $\mathbf{Cat}_{\mathrm{Ladj}}^{\mathrm{op,co}}$ of categories with right adjoint functors and natural transformations, respectively, of left adjoint functors and natural transformations with both 1- and 2-morphisms reversed.

Now a monad in a 2-category $\mathcal K$ is equivalently a monad in $\mathcal K^{\mathrm{op}}$ (a monoid in a monoidal category remains a monoid when the monoidal product is reversed) and a comonad in $\mathcal K^{\mathrm{co}}$ (a monoid in a monoidal category is a comonoid in the opposite monoidal category.)

Thus a monad $(T,\mu,\eta)$ on a category $\mathcal C$ uniquely determines a comonad structure $(U,\bar\mu,\bar\eta)$ on any left or right adjoint to $T$, subject to the condition that the comonad structure should be related to the monad structure via the calculus of mates.

EDIT: I see I missed the point of the question, which is whether there is a unique comonad structure on the left or right adjoint of a monad. There is no reason for this at all. The paper you link should assume that the given comonad is adjoint to the given monad and that the comonad structure is given via mates as above. An easy example of a functor with a right adjoint and multiple monad structures is $M\times(-)$, for any set $M$ admitting multiple monoid structures. For a functor with a right adjoint and multiple comonad structures, consider a vector space $V$ with multiple coalgebra structures. This puts multiple comonad structures on the endofunctor $V\otimes (-)$ of vector spaces, which is a left adjoint.